Comparative statistical analysis of the release kinetics models for nanoprecipitated drug delivery systems based on poly(lactic-co-glycolic acid)

Poly(lactic-co-glycolic acid) is one of the most used polymers for drug delivery systems (DDSs). It shows excellent biocompatibility, biodegradability, and allows spatio-temporal control of the release of a drug by altering its chemistry. In spite of this, few formulations have reached the market. To characterize and optimize the drug release process, mathematical models offer a good alternative as they allow interpreting and predicting experimental findings, saving time and money. However, there is no general model that describes all types of drug release of polymeric DDSs. This study aims to perform a statistical comparison of several mathematical models commonly used in order to find which of them best describes the drug release profile from PLGA particles synthesized by nanoprecipitation method. For this purpose, 40 datasets extracted from scientific articles published since 2016 were collected. Each set was fitted by the models: order zero to fifth order polynomials, Korsmeyer-Peppas, Weibull and Hyperbolic Tangent Function. Some data sets had few observations that do not allow to apply statistic test, thus bootstrap resampling technique was performed. Statistic evidence showed that Hyperbolic Tangent Function model is the one that best fit most of the data.


Response:
The authors thank this valuable comment. We considered 40 datasets extracted from scientific literature since 2016. This is due to three main reasons listed below.
1. Using the nanoprecipitation method there are hundreds of scientific papers published so far, therefore, to limit the study, the papers from the last 5 years, i.e. since 2016, were selected. 2. In 2016, to the best of our knowledge, a paper referring to the hyperbolic tangent function model was published for the first time (Eltayeb, Stride, Edirisinghe, & Harker, 2016). 3. The authors considered the time of 5 years as sufficient for other authors to consider and use the hyperbolic tangent model and thus collect representative data and results.
For a better understanding of the readers an explanation of this was added in lines 170-172 of the manuscript where it is expressed: To limit the study, the authors selected scientific articles from 2016. Furthermore, to the best of our knowledge, this was the year in which the mathematical model of hyperbolic tangent function was applied for the first time to evaluate the release profile of core-shell lipid nanoparticles [77].

Response:
Accordingly to this comment, we have added the following schematization to address point i) as figure #2.
Here the authors show a timeline of the development of the models used in the present study.
In the manuscript, line 200-203 introduce the reader to the aforementioned timeline: 3. Few English sentences are confusing. I will suggest to check/correct the whole manuscript by a native

Response:
Thank you for your suggestion, whole manuscript has been revised by a Native American. All changes are recorded using Track Changes in Microsoft Word

Response:
Thank you for your suggestion. We have moved Table 1 and 2 to SI. These tables were selected because they are not main results. Table 1 corresponds to standard equations and Table 2 to scientifical well-known definitions and general equations.
It should be noted that the numbers of the figures and tables in the manuscript and in the SI have changed, these have been updated and also their citations within the text.

Response:
Thank you for pointing this out. Conclusion has been rewritten as follow (lines 524-541): This study demonstrated that, although they are tools that help to understand drug release dynamics, there is no general empirical/semiempirical mathematical expression that can describe, in all cases, the release profile of drugs encapsulated in PLGA nanoparticles synthesized by nanoprecipitation methods. In addition, it was found that although R 2 and R 2 a are the most commonly used criteria to determine whether or not a model fits the data obtained in an experiment, they are not the most appropriate. It was also shown that the Akaike and Bayesian criteria can better reflect the fit of a model since the results are not influenced by the inclusion of new terms or an increase in the complexity of the equation. It was also revealed that the number of observations per set is a limiting factor for the application of different models and the subsequent statistical analysis. Therefore, the Bootstrap resampling technique becomes a very useful technique to solve this drawback. In the specific case of this study, 50% of the sets studied do not meet the requirement of at least 8 observations for the construction of the release curve, therefore these results are not very reproducible and have low statistical significance.
Furthermore, the analysis employed in this project provided significant statistical evidence to consider the Hyperbolic Tangent Function model as the most adequate and general model to describe the drug release kinetics. This model, unlike the mathematical expression of Korsmeyer-Peppas, could be adjusted to the complete release curve. Since it has kinetic parameters, it acquires greater predictive power than the Weibull model. However, a more exhaustive study of this model is required in order to understand the chemistry, physics, and biology behind it.

Responses to the comments of Reviewer #2
Reviewer #2: The manuscript applied the appropriate statistical approaches to compare and summarize the commonly used models for the drug release profile from PLGA particles synthesized by nano-precipitation method. The screening and justification of literature and data are important for the assessment of study outcomes and final conclusion. It will be better to add a flow diagram for the screening procedure and justification/criteria of literature for the systematic research.

Response:
We think this is an excellent suggestion, therefore, we have added Fig. 1 and its citation in the manuscript.
For a better understanding of the readers an explanation of the selection criteria was added in lines 170-172 of the manuscript where it is expressed: To limit the study, the authors selected scientific articles from 2016. Furthermore, to the best of our knowledge, this was the year in which the mathematical model of hyperbolic tangent function was applied for the first time to evaluate the release profile of core-shell lipid nanoparticles [77].